Subgroups — #36374055

Order 22336.

202 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19150210	C₂	1	3	yes	maximal
2	#26013546	C₂	1	3	yes	maximal, complemented
3	C₂²×Q₈×C₃₄₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂³×D₃₄₉	C₂²	1	1	yes	characteristic
5	C₂²×Dic₃₄₉	C₂²	1	3	yes	—
6	#19150210	—	2	6	no	—
7	C₂×Q₈×C₃₄₉	—	2	6	no	—
8	C₂²×C₄×C₃₄₉	C₂²	1	3	yes	—
9	C₂²×D₃₄₉	—	2	3	no	—
10	C₂²×D₃₄₉	—	2	3	no	—
11	C₂²×D₃₄₉	Q₈	1	2	yes	complemented
12	C₂×Dic₃₄₉	—	2	3	no	—
13	C₂×Dic₃₄₉	—	2	6	no	—
14	C₂³×C₃₄₉	C₂³	1	1	yes	characteristic
15	Q₈×C₃₄₉	—	2	8	no	—
16	C₂×C₄×C₃₄₉	—	2	6	no	—
17	C₂×C₄×C₃₄₉	D₄	1	6	yes	—
18	C₂²×C₃₄₉	C₂×D₄	1	3	yes	—
19	C₂²×C₃₄₉	C₂×Q₈	1	1	yes	characteristic
20	C₂²×C₃₄₉	#41	1	3	yes	—
21	C₂×D₃₄₉	—	2	6	no	—
22	C₂×D₃₄₉	—	4	1	no	—
23	C₂×D₃₄₉	—	4	3	no	—
24	C₄×C₃₄₉	—	2	6	no	—
25	Dic₃₄₉	—	4	3	no	—
26	C₂×C₃₄₉	#120	1	1	yes	characteristic
27	C₂×C₃₄₉	C₂²⋊Q₈	1	3	yes	—
28	C₂×C₃₄₉	#124	1	3	yes	—
29	D₃₄₉	—	4	2	no	—
30	C₃₄₉	#393	1	1	yes	characteristic, complemented, Sylow
31	#393	—	349	1	no	maximal, Sylow
32	C₂²×Q₈	D₃₄₉	1	1	yes	characteristic, complemented
33	C₂×C₂²⋊C₄	—	349	3	no	—
34	#95	—	349	3	no	—
35	C₂⁴	—	349	1	no	—
36	C₂×Q₈	—	2	6	no	—
37	C₂²×C₄	—	349	3	no	—
38	C₂²×C₄	C₂×D₃₄₉	1	3	yes	—
39	C₂²⋊C₄	—	698	6	no	—
40	C₂³	—	349	2	no	—
41	C₂³	—	698	3	no	—
42	C₂³	—	698	3	no	—
43	C₂³	C₂²×D₃₄₉	1	1	yes	characteristic
44	Q₈	—	2	8	no	—
45	C₂×C₄	—	2	6	no	—
46	C₂×C₄	—	698	3	no	—
47	C₂×C₄	—	698	6	no	—
48	C₂×C₄	C₂²⋊D₃₄₉	1	6	yes	—
49	C₂²	—	1396	1	no	—
50	C₂²	—	1396	3	no	—
51	C₂²	—	698	6	no	—
52	C₂²	C₂×C₂²⋊D₃₄₉	1	3	yes	—
53	C₂²	Q₈×D₃₄₉	1	1	yes	characteristic
54	C₂²	#19150234	1	3	yes	—
55	C₄	—	2	6	no	—
56	C₄	—	1396	3	no	—
57	C₂	—	1396	2	no	—
58	C₂	#26013608	1	1	yes	characteristic
59	C₂	#26013615	1	3	yes	—
60	C₂	#26013628	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₃₄₉
commutator, socle, cosocle	C₂³×C₃₄₉	C₂³
Fitting	C₂²×Q₈×C₃₄₉	C₂
prime radical	C₂³×D₃₄₉	C₂²
hypercenter	C₂²×Q₈	D₃₄₉
hypocenter	C₃₄₉	#393
LCS	G ▷ C₂³×C₃₄₉ ▷ C₃₄₉
UCS	C₁ ◁ C₂³ ◁ C₂²×Q₈

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. #26013546	C₂	semidirect
3. C₂²×Q₈×C₃₄₉	C₂	semidirect
11. C₂²×D₃₄₉	Q₈	semidirect
30. C₃₄₉	#393	semidirect
32. C₂²×Q₈	D₃₄₉	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#19150210	2
#26013546	2
C₂²×Q₈×C₃₄₉	2
#393	349